Question: Perform the row operation, $R_2 \leftrightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} -10 & 2 & -8 & 1 \\ 3 & -2 & 2 & 2 \\ 1 & 4 & -6 & 2 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Switching the rows For the given matrix, $R_2$ and $R_3$ are given below. $R_2=\left[\begin{array} {ccc} 3 & -2 & 2 & 2 \end{array} \right] ~~~~~ R_3=\left[\begin{array} {ccc} 1 & 4 & -6 & 2 \end{array} \right]$ $\left[\begin{array} {ccc} -10 & 2 & -8 & 1 \\ 3 & -2 & 2 & 2 \\ 1 & 4 & -6 & 2 \end{array} \right]\xrightarrow{R_2 \leftrightarrow R_3}\left[\begin{array} {ccc} -10 & 2 & -8 & 1 \\ 1 & 4 & -6 & 2 \\ 3 & -2 & 2 & 2 \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -10 & 2 & -8 & 1 \\ 1 & 4 & -6 & 2 \\ 3 & -2 & 2 & 2 \end{array} \right]$